# Leggett–Garg-like Inequalities from a Correlation Matrix Construction

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

**Theorem 1.**

**Proof of Theorem 1.**

**Theorem 2.**

**Proof of Theorem 2.**

**Theorem 3.**

**Proof of Theorem 3.**

**Theorem 4.**

**Proof of Theorem 4.**

#### An Example of a System That Upholds Our New Bounds

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Leggett, A.J.; Garg, A. Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? Phys. Rev. Lett.
**1985**, 54, 857–860. [Google Scholar] [CrossRef] - Emary, C.; Lambert, N.; Nori, F. Leggett-Garg inequalities. Rep. Prog. Phys.
**2013**, 77, 016001. [Google Scholar] [CrossRef] - Majidy, S.S.; Katiyar, H.; Anikeeva, G.; Halliwell, J.; Laflamme, R. Exploration of an augmented set of Leggett-Garg inequalities using a noninvasive continuous-in-time velocity measurement. Phys. Rev. A
**2019**, 100, 042325. [Google Scholar] [CrossRef] - Majidy, S.; Halliwell, J.J.; Laflamme, R. Detecting violations of macrorealism when the original Leggett-Garg inequalities are satisfied. Phys. Rev. A
**2021**, 103, 062212. [Google Scholar] [CrossRef] - Knee, G.C.; Simmons, S.; Gauger, E.M.; Morton, J.J.; Riemann, H.; Abrosimov, N.V.; Becker, P.; Pohl, H.J.; Itoh, K.M.; Thewalt, M.L.; et al. Violation of a Leggett–Garg inequality with ideal non-invasive measurements. Nat. Commun.
**2012**, 3, 606. [Google Scholar] [CrossRef] - Shenoy, A.; Aravinda, S.; Srikanth, R.; Home, D. Can the use of the Leggett-Garg inequality enhance security of the BB84 protocol? Phys. Lett. A
**2017**, 381, 2478–2482. [Google Scholar] [CrossRef] - Bennett, C.H.; Brassard, G. Quantum cryptography: Public key distribution and coin tossing. arXiv
**2020**, arXiv:2003.06557. [Google Scholar] [CrossRef] - Bell, J.S. On the Einstein Podolsky Rosen paradox. Phys. Phys. Fiz.
**1964**, 1, 195. [Google Scholar] [CrossRef] - Clauser, J.F.; Horne, M.A.; Shimony, A.; Holt, R.A. Proposed Experiment to Test Local Hidden-Variable Theories. Phys. Rev. Lett.
**1969**, 23, 880–884. [Google Scholar] [CrossRef] - Cirel’son, B.S. Quantum generalizations of Bell’s inequality. Lett. Math. Phys.
**1980**, 4, 93–100. [Google Scholar] [CrossRef] - Budroni, C.; Moroder, T.; Kleinmann, M.; Gühne, O. Bounding temporal quantum correlations. Phys. Rev. Lett.
**2013**, 111, 020403. [Google Scholar] [CrossRef] - Fritz, T. Quantum correlations in the temporal Clauser–Horne–Shimony–Holt (CHSH) scenario. New J. Phys.
**2010**, 12, 083055. [Google Scholar] [CrossRef] - Carmi, A.; Cohen, E. On the significance of the quantum mechanical covariance matrix. Entropy
**2018**, 20, 500. [Google Scholar] [CrossRef] - Carmi, A.; Cohen, E. Relativistic independence bounds nonlocality. Sci. Adv.
**2019**, 5, eaav8370. [Google Scholar] [CrossRef] - Carmi, A.; Herasymenko, Y.; Cohen, E.; Snizhko, K. Bounds on nonlocal correlations in the presence of signaling and their application to topological zero modes. New J. Phys.
**2019**, 21, 073032. [Google Scholar] [CrossRef] - Te’eni, A.; Peled, B.Y.; Cohen, E.; Carmi, A. Multiplicative Bell inequalities. Phys. Rev. A
**2019**, 99, 040102. [Google Scholar] [CrossRef] - Cohen, E.; Carmi, A. In praise of quantum uncertainty. Entropy
**2020**, 22, 302. [Google Scholar] [CrossRef] - Fröwis, F.; Sekatski, P.; Dür, W. Detecting large quantum Fisher information with finite measurement precision. Phys. Rev. Lett.
**2016**, 116, 090801. [Google Scholar] [CrossRef] - Robertson, H.P. The uncertainty principle. Phys. Rev.
**1929**, 34, 163. [Google Scholar] [CrossRef] - Schrödinger, E. Zum Heisenbergschen Unschärfeprinzip; Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse: Berlin, Germany, 1930; Volume 14, pp. 230–296. [Google Scholar]
- Adesso, G.; Illuminati, F. Entanglement in continuous-variable systems: Recent advances and current perspectives. J. Phys. A Math. Theor.
**2007**, 40, 7821. [Google Scholar] [CrossRef] - Pirandola, S.; Serafini, A.; Lloyd, S. Correlation matrices of two-mode bosonic systems. Phys. Rev. A
**2009**, 79, 052327. [Google Scholar] [CrossRef] - Tsirel’son, B.S. Quantum analogues of the Bell inequalities. The case of two spatially separated domains. J. Sov. Math.
**1987**, 36, 557–570. [Google Scholar] [CrossRef] - Landau, L.J. Empirical two-point correlation functions. Found. Phys.
**1988**, 18, 449–460. [Google Scholar] [CrossRef] - Masanes, L. Necessary and sufficient condition for quantum-generated correlations. arXiv
**2003**, arXiv:quant-ph/0309137. [Google Scholar] - Dressel, J.; Korotkov, A.N. Avoiding loopholes with hybrid Bell-Leggett-Garg inequalities. Phys. Rev. A
**2014**, 89, 012125. [Google Scholar] [CrossRef] - White, T.C.; Mutus, J.Y.; Dressel, J.; Kelly, J.; Barends, R.; Jeffrey, E.; Sank, D.; Megrant, A.; Campbell, B.; Chen, Y.; et al. Preserving entanglement during weak measurement demonstrated with a violation of the Bell–Leggett–Garg inequality. Npj Quantum Inf.
**2016**, 2, 15022. [Google Scholar] [CrossRef] - Halliwell, J. Leggett-Garg inequalities and no-signaling in time: A quasiprobability approach. Phys. Rev. A
**2016**, 93, 022123. [Google Scholar] [CrossRef] - Vitale, V.; De Filippis, G.; De Candia, A.; Tagliacozzo, A.; Cataudella, V.; Lucignano, P. Assessing the quantumness of the annealing dynamics via Leggett Garg’s inequalities: A weak measurement approach. Sci. Rep.
**2019**, 9, 13624. [Google Scholar] [CrossRef] - Santini, A.; Vitale, V. Experimental violations of Leggett-Garg inequalities on a quantum computer. Phys. Rev. A
**2022**, 105, 032610. [Google Scholar] [CrossRef] - Carmi, A.; Moskovich, D. Tsirelson’s bound prohibits communication through a disconnected channel. Entropy
**2018**, 20, 151. [Google Scholar] [CrossRef] - Liu, J.; Yuan, H.; Lu, X.M.; Wang, X. Quantum Fisher information matrix and multiparameter estimation. J. Phys. Math. Theor.
**2020**, 53, 023001. [Google Scholar] [CrossRef] - Athalye, V.; Roy, S.S.; Mahesh, T. Investigation of the Leggett-Garg inequality for precessing nuclear spins. Phys. Rev. Lett.
**2011**, 107, 130402. [Google Scholar] [CrossRef] [PubMed] - Aharonov, Y.; Albert, D.Z.; Vaidman, L. How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett.
**1988**, 60, 1351–1354. [Google Scholar] [CrossRef] - Moiseyev, N. Non-Hermitian Quantum Mechanics; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Budroni, C.; Emary, C. Temporal quantum correlations and Leggett-Garg inequalities in multilevel systems. Phys. Rev. Lett.
**2014**, 113, 050401. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**

**Demonstration of Theorems 1–3 for our spin model**. (

**a**,

**b**), the values of $D1$ and $D2$, respectively, both of which have a minimal value of 0, indicating Equations (3) and (13) are upheld. (

**c**) The indigo area represents all data points, while the grey area represents the bounds of half of the unit sphere. All indigo points are within the gray area, indicating Equation (17) is upheld.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ben Porath, D.; Cohen, E.
Leggett–Garg-like Inequalities from a Correlation Matrix Construction. *Quantum Rep.* **2023**, *5*, 398-406.
https://doi.org/10.3390/quantum5020025

**AMA Style**

Ben Porath D, Cohen E.
Leggett–Garg-like Inequalities from a Correlation Matrix Construction. *Quantum Reports*. 2023; 5(2):398-406.
https://doi.org/10.3390/quantum5020025

**Chicago/Turabian Style**

Ben Porath, Dana, and Eliahu Cohen.
2023. "Leggett–Garg-like Inequalities from a Correlation Matrix Construction" *Quantum Reports* 5, no. 2: 398-406.
https://doi.org/10.3390/quantum5020025